If that were my budget/requirements, I might look around Taikoo Shing or Sai Wan Ho area - estates like Grand Promenade or Les Saisons for example. Although it is a little further from Central, if that is important to you. But it's very close to Quarry Bay and quite convenient as well.

How important is being close to the MTR to you? Because there are many places up the hill a bit which would fit your requirements - some on North Point Hill (such as Pacific Palisades) but I find that once you're up the hill, many times the proximity to shops is not so good - although since you're higher, the light will be better. There are also places up the hill in Tai Hung, Wan Chai or even Central Mid Levels - but as you go further west, the commute times increase.

I live in Tai Tam (near Stanley) and it's a quick 15-20 minute commute to Quarry Bay. The area is not MTR accessible, but many of the estates around here such as Pacific View, Redhill Peninsula and Manhattan provide free shuttle buses regularly to MTR stations. Tai Tam or even Stanley is nice because it feels suburban and takes you completely out of the city if you ever need refuge from busy HK. With your budget, you could easily find a spacious 2-3 bedroom apartment with sea view and club house facilities.

Additionally, if you plan to drive at all, then the commute can become quite easy. I never hit any traffic going from Tai Tam/Stanley into Eastern HK.

Or Clearwater Bay is lovely. Some areas are 30 minutes by car to Quarry Bay or if you live near Hang Hau you can jump on the MTR and it is only 15 minutes or so. You'd get much more space there for your money, and lovely scenery. Not sure about the 'modern building w/pool' though.

If not I second evgreen. I used to live in Shek O (further than Tai Tam) and commute to Quarry Bay. Used to take me 20-30 minutes (by car).

I've been looking tonight at the Stanley area, as evgreen mentioned. It looks really nice, with a village atmosphere.
But I am not sure we can get a 3 bed place here on our budget?
Seems to be a nice area where we can push a pram around, with some cafes/bars etc.
I know there is no MTR here - do buses run from here or do people drive/taxi to central?

Stanley is lovely. A great supermarket, good bars and restaurants. You might be able to find something on your budget but you'll have to search I suspect. It is a really nice place though. Buses run from Stanley. Taxis will too, but it can be tough to get a taxi in Hong Kong at certain times. If I were you I would definitely get a car. Especially if you have two little ones. It'll be much more convenient and a lot safer also.

I've been looking tonight at the Stanley area, as evgreen mentioned. It looks really nice, with a village atmosphere.
But I am not sure we can get a 3 bed place here on our budget?
Seems to be a nice area where we can push a pram around, with some cafes/bars etc.
I know there is no MTR here - do buses run from here or do people drive/taxi to central?

Thanks for your all your time to respond!

Public transport from Stanley is ultra convenient. Several bus routes available going to Causeway Bay, Chai Wan, Central and Aberdeen. I would look into Stanford Villa in Stanley if you're looking at a 3 bedroom on your budget. There is a gym and pool on the property. Some apartment units there even have patios/rooftops available. I used to live there with my husband and it's an easy walk to several restaurants, 7-11, supermarket and even the beach.

Ummm, unless you are at the beginning of the ride, minibuses get full very quickly coming out of Stanley. In fact:
1) without a car, living anywhere in the southside is not particularly convenient for rush hour commute or even for popping out for errands in mid day
2) with a car, you have the issues of running a car on Hong Kong which are numerous.

Without actually being here, you wont know what you will like and what your family will tolerate. You are lucky that you have a fairly generous budget. I would suggest you come and stay somewhere temporary for a month or 2 before deciding.

I would recommend QuarryBay/Taikoo (walking distance) followed by Shaukeiwan and SiuSaiWan (15 mins). Shops, MTR with pools and seaviews. Ticks all the boxes.

Clearwaterbay, ShekO, Stanley, TaiTam are all beautiful areas but every single one of them has traffic issues as they are on single lane (each way) roads. One small accident and you are stuck for a while. That sucks when you have to get to work quickly.

When you do come here, also consider Happy Valley. Its a happy place, will tick all boxes except for the seaview and makes a very centrally located place that is convenient to everywhere.